Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit

We consider the behaviour of the distribution for stationary solutions of the complex Ginzburg-Landau equation perturbed by a random force. It was proved earlier that if the random force is proportional to the square root of the viscosity, then the family of stationary measures possesses an accumulation point as the viscosity goes to zero. We show that if $\mu$ is such point, then the distributions of the L^2 norm and of the energy possess a density with respect to the Lebesgue measure. The proofs are based on It\^o's formula and some properties of local time for semimartingales.Comment: 12 page

Quantum ring models and action-angle variables

We suggest to use the action-angle variables for the study of properties of (quasi)particles in quantum rings. For this purpose we present the action-angle variables for three two-dimensional singular oscillator systems. The first one is the usual (Euclidean) singular oscillator, which plays the role of the confinement potential for the quantum ring. We also propose two singular spherical oscillator models for the role of the confinement system for the spherical ring. The first one is based on the standard Higgs oscillator potential. We show that, in spite of the presence of a hidden symmetry, it is not convenient for the study of the system's behaviour in a magnetic field. The second model is based on the so-called CP(1) oscillator potential and respects the inclusion of a constant magnetic field.Comment: 9 pages, nofigure